Conjecture on the asymptotic second eigenvalue of the Schoenberg operator

Let $\gamma_{\Delta_n,k}$ denote the second largest eigenvalue of the Schoenberg operator associated with the fixed grid $\Delta_n$ and spline degree $k$.

Schoenberg eigenvalue conjecture. For fixed $k>0$,

$$\gamma_{\Delta_n,k}\to 1\quad\text{as }n\to\infty.$$

For fixed $n>0$,

$$\gamma_{\Delta_n,k}\to 1\quad\text{as }k\to\infty.$$

The conjecture concerns whether the second largest eigenvalue can affect the convergence rate in the lower approximation bound. The source states that the eigenvalues and eigenfunctions of the Schoenberg operator are otherwise unknown and provides no resolution.